Im trying to get my head around complex integration/complex line integrals. Real integration can be thought of as the area under a curve or the opposite of differentiation. Thinking of it geometrically as the area under a curve or the volume under a surface in 3 dimension is very intuitive.
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So is there a geometric way of thinking about complex integration? Or should I just be viewing it as process that reverses differentiation? Or has integration other meanings in complex analysis?
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Here is an example. Could someone explain this to me?
Here's the definition of the integral along a curve gamma in $\mathbb{C}$, parameterized by $w(t):[a, b] \to \mathbb{C}$.
$$\int_\gamma f(z) \mathrm{d}z = \int_a^b f[w(t)] w'(t) \mathrm{d}t$$
So I have that:
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$\gamma$ is the unit circle with anti-clockwise orientation parameterized by $w:[0, 2\pi] \to \mathbb{C}$.
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$w(t) = e^{it} = \cos(t) + i \sin(t)$.
So if we use the definition of the integral,
$$\int_\gamma f(z) \mathrm{d}z = \int^b_a f[w(t)]w'(t) \mathrm{d}t$$
and work this out, it comes to
$$\int^{2\pi}_0 i \mathrm{d}t = 2{\pi}i$$
So what does this $2{\pi}i$ represent? Does it mean anything geometrically, like if a regular integral works out to be 10 that means the area under the curve between 2 points is 10…
Best Answer
In every book or course where line integrals occur (in physics, complex analysis, geometry, $\ldots$) people try to explain in so many (english!) words what a line integral is. It's not always the same thing and comes in various forms. As a rule of thumb one can say the following:
A line integral is a function that assigns to any (real scalar, complex scalar, vector) field $f$ on a domain $\Omega$ and any (directed) curve $\gamma\subset\Omega$ a certain value $v$, denoted by $$\int_\gamma f(x)* dx$$ (or similar). This rule should have the following properties; the first one giving the geometric or physical intuition behind $v$:
When $f$ is constant and $\gamma$ is the segment with initial point $x_0$ and endpoint $x_1$ then $v=f*(x_1-x_0)$.
The value $v$ is independent of the chosen parametrization of $\gamma$.
When $\gamma=\gamma_1+\gamma_2$ in an obvious way then $$\int_\gamma f(x)* dx =\int_{\gamma_1} f(x)* dx +\int_{\gamma_2} f(x)* dx\ .$$
Here $*$ denotes any multiplication that makes sense in the actual situation, and $dx$ might as well be $|dx|$ in certain cases.